How to cut out unwanted content from the 3D model
Sometimes it can save time to build the new geometry you need.



In this case:


Add a new UV sphere aligned to the world axes, with twice as many segments as rings. (Here, 32 and 16)
Delete the exterior faces over the section to be removed.
Put your snap settings to vertex, active, snap to self.
Extrude the vertex at the north pole downwards, constraining to Z, (E,Z) and snapping to a convenient vertex on the equator. This will create the edge from the north pole to the centre of the sphere.
Set 'Automatically merge vertices' on, and extrude the new central vertex outwards twice, once to each of the bottom corners of the removed section of the sphere. This will create the other two straight edges you can see.
Select all the new straight edges, and from the (ctrl-E) edges menu, select subdivide. At the bottom of the Tools panel, set the number of cuts to 3.
Fer each new flat face you are going to create, select the edges surrounding it, go to the face menu (ctrl-F) and select 'Grid Fill'. At the bottom of the tools panel, set the span to 4.
Bevel the edges of your new faces, if you want to.


It may all seem like a bit of a faff, but once you can find your way around, this sort of approach can take less time than tidying up after a Boolean. (I'm sure someone here will point out how I could have saved a few steps.)